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Theorem iunsuc 4310
Description: Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
Hypotheses
Ref Expression
iunsuc.1  |-  A  e. 
_V
iunsuc.2  |-  ( x  =  A  ->  B  =  C )
Assertion
Ref Expression
iunsuc  |-  U_ x  e.  suc  A B  =  ( U_ x  e.  A  B  u.  C
)
Distinct variable groups:    x, A    x, C
Allowed substitution hint:    B( x)

Proof of Theorem iunsuc
StepHypRef Expression
1 df-suc 4261 . . 3  |-  suc  A  =  ( A  u.  { A } )
2 iuneq1 3794 . . 3  |-  ( suc 
A  =  ( A  u.  { A }
)  ->  U_ x  e. 
suc  A B  = 
U_ x  e.  ( A  u.  { A } ) B )
31, 2ax-mp 5 . 2  |-  U_ x  e.  suc  A B  = 
U_ x  e.  ( A  u.  { A } ) B
4 iunxun 3860 . 2  |-  U_ x  e.  ( A  u.  { A } ) B  =  ( U_ x  e.  A  B  u.  U_ x  e.  { A } B )
5 iunsuc.1 . . . 4  |-  A  e. 
_V
6 iunsuc.2 . . . 4  |-  ( x  =  A  ->  B  =  C )
75, 6iunxsn 3857 . . 3  |-  U_ x  e.  { A } B  =  C
87uneq2i 3195 . 2  |-  ( U_ x  e.  A  B  u.  U_ x  e.  { A } B )  =  ( U_ x  e.  A  B  u.  C
)
93, 4, 83eqtri 2140 1  |-  U_ x  e.  suc  A B  =  ( U_ x  e.  A  B  u.  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    e. wcel 1463   _Vcvv 2658    u. cun 3037   {csn 3495   U_ciun 3781   suc csuc 4255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-iun 3783  df-suc 4261
This theorem is referenced by: (None)
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